Foundation Electron Mass From Phi Ladder Electron Muon Ratio Rs Pos

The electron-muon ratio

The electron and the muon are two of the charged leptons, the family of particles that also includes the tau. The muon is heavier than the electron: the measured ratio of their masses is 206.77. The question is whether that number has a simple origin.

In Recognition Science, the framework models particle masses as powers of the golden ratio phi, about 1.618, arranged on a ladder of integer rungs. The electron sits at rung 8, the muon at rung 14, a gap of 6 rungs. The framework's declaration electron_muon_ratio_RS_pos is a proved theorem stating that the ratio it defines, phi to the sixth power, is positive. A companion theorem pins its value between 17.9 and 18.0.

That value is far from the measured 206.77. The framework's own structural falsifier states the condition that would break the ladder: a precision measurement placing the electron-to-muon ratio outside any integer power of phi from phi^1 through phi^12 by more than about 0.118 on a log-mass scale. The current measured value sits between phi^11, about 199.0, and phi^12, about 321.8, with the nearest power phi^11 about 3.8 percent off.

What the declaration does not claim is that the framework has derived the true electron-muon mass ratio. It establishes an internal, structural quantity, a definitional choice within the framework's model, together with its positivity and numerical band. The dimensional bridge that would select the correct power of phi from the measured value is named as a follow-on problem, not solved here.

theorem electron_muon_ratio_RS_pos : 0 < electron_muon_ratio_RS :=
  pow_pos phi_pos 6

/-- Numerical band: `phi^6 ∈ (17.9, 18.0)`.
`phi^6 = (phi^3)^2 = (2 phi + 1)^2 = 4 phi^2 + 4 phi + 1
  = 4(phi + 1) + 4 phi + 1 = 8 phi + 5`.
With `1.61 < phi < 1.62`, `17.88 < 8 phi + 5 < 17.96`. -/
theorem electron_muon_ratio_RS_band :
    17.9 < electron_muon_ratio_RS ∧ electron_muon_ratio_RS < 18.0 := by
  unfold electron_muon_ratio_RS
  have h1 := phi_gt_onePointSixOne
  have h2 := phi_lt_onePointSixTwo
  have hsq := phi_sq_eq
  have : phi ^ (6 : ℕ) = (phi ^ (3 : ℕ)) ^ 2 := by ring
  rw [this]
  have hcube : phi ^ (3 : ℕ) = phi * (phi + 1) := by
    have : phi ^ (3 : ℕ) = phi * phi ^ 2 := by ring
    rw [this, hsq]
  rw [hcube]
  refine ⟨?_, ?_⟩ <;> nlinarith [hsq]

theorem electron_muon_ratio_RS_pos : 0 < electron_muon_ratio_RS :=
  pow_pos phi_pos 6

What this page does not claim

The framework has not derived the measured electron-muon mass ratio of 206.77. The declaration does not claim the phi^6 value is a prediction matching experiment. The framework's model does not yet identify the dimensional scale that connects its native units to conventional physics.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/ElectronMassFromPhiLadder.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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